The woman, in her 40s, recently arrived in Victoria where she spent two weeks in quarantine and returned two negative results for the coronavirus.

She flew into Adelaide on Sunday night and subsequently tested positive.

Premier Steven Marshall says the result is a "low positive" and the woman is not considered infectious.

However, she will spend another two weeks in self-isolation along with the rest of her family.

Other family members have tested negative.

Deputy Chief Public Health Officer Mike Cusack says the woman did all the right things, presenting for an immediate test upon arriving in Adelaide.

The initial swab showed extremely low levels of the virus, as did a second test which confirmed her infection.

The results were an indication the woman was in the "recovery phase" of the disease, Dr Cusack said.

"As part of the test, we have to amplify the virus and we were almost at the testing limit," he said.

"We had to do approaching 40 amplifications just to detect any particles of virus or any fragments of virus."

Dr Cusack said an antibody test subsequently showed the infection had probably occurred in the previous two weeks or so.

"They've still got a little bit of virus shedding but they're certainly not infectious and they're not a risk to the public," he said.

The woman's case takes the SA total since the start of the pandemic to 444.

SA has also decided to hit anyone who enters the state from Victoria with a $1000 fine if they fail to get tested within 24 hours.

The new measure will take effect from midnight on Saturday night.

People allowed into SA from Victoria could also be asked to pay for the cost of their 14-day quarantine.

Health Minister Stephen Wade said the costs would not be passed on as a matter of course but would be considered on a case-by-case basis.

Australian Associated Press